Date: Mar 15, 2013 6:06 PM
Subject: Re: Matheology � 224
William Hughes <firstname.lastname@example.org> wrote:
> On Mar 14, 10:32 am, WM <mueck...@rz.fh-augsburg.de> wrote:
> >... consider the list of finite initial segments of natural numbers
> > 1
> > 1, 2
> > 1, 2, 3
> > ...
> > According to set theory it contains all aleph_0 natural numbers in its
> > lines. But is does not contain a line containing all natural numbers.
> > Therefore it must be claimed that more than one line is required to
> > contain all natural numbers. This means at least two line are
> > necessary. There are no special lines necessary, but there must be at
> > least two. In this case, however, we can prove, by the construction of
> > the list, that every union of a pair of lines is contained in one of
> > the lines. This contradicts the assertion that all natural numbers
> > exist and are in lines of the list.
> Nope, two lines are necessary but not sufficient.
> Two lines can never do a better job than 1.
> Any finite number of lines is necessary but not sufficient.
> Any finite number of lines can never do a better job than 1.
> An infinite number of lines is necessary and sufficient
> An infinite number of lines can do a better job than 1
> [In potential infinity things go
> Any number of findable lines is not sufficient
> An unfindable line is necessary and sufficient
> An unfindable line can do a better job than
> any number of findable lines.
Or it would if you could find it.